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# Geographic Information Science Modeling Applications GEOG 4203

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This 33 page Class Notes was uploaded by Jeremy Steuber on Thursday October 29, 2015. The Class Notes belongs to GEOG 4203 at University of Colorado at Boulder taught by Stefan Leyk in Fall. Since its upload, it has received 18 views. For similar materials see /class/231898/geog-4203-university-of-colorado-at-boulder in Geography at University of Colorado at Boulder.

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Date Created: 10/29/15

Geography 4203 5203 GIS Modeling Class Block 9 Variogram amp Kriging Some Updates Today class one proposal presentation Feb 22 Proposal Presentations Feb 25 Readings discussion Interpolation Last Lecture You have seen the first part of spatial estimation which was about interpolation techniques You hopefully obtained an idea of what the basic idea of interpolation is and when and why me are supposed to make use of it You had some insights into the concepts of different techniques such as NN pycnophylactiv l IDW Splines You hopefully understood the mathematical foundation of these approaches to better understand what you are doing in the lab Today s Outline We will begin with Geostatistics which means we talk about Kriging It is important to understand the difference between Kriging and interpolation techniques we had so far We will talk about the conceptual basics constraints and methods for Kriging without going too much into detail Here we need a deeper understanding of autocorrelation and regionalized variable theory as well as the incorporation of semivariogram models into estimations Learning Objectives You will understand how Kriging works and why it is called an optimized local interpolator You will better understand terms like autocorrelation semivariance covariance You will understand the mathematical foundation of Kriging and how we make use of spatial autocorrelation and semivariance to estimate weights for estimating Back to How to Improve Spatial Estimation Remember how we incorporated autocorrelation into interpolation so far lmplicitly we just assumed the data show spatial dependence we knew this from data exploration Parameter settings eg in IDW are arbitrary Radius neighbors Weighting we get more knowledge through testing different combinations How to make theorybased choices and parameter settings This is where Geostatistical methods come into the picture to help us with this decisionmaking process Predictions can be improved by incorporating the knowledge of autocorrelation knowing the value at one location provides information of locations nearby Geostatistics Too many definitions to go into detail Basically it is about the analysis and inference of continuouslydistributed variables temperature concentrations Analysis describing the spatial variability of the phenomenon under study Inference Estimation of the values at unknown locations unknown values Overall we need techniques for statistical estimation of spatial phenomena and this is what Geostatistics provides Tobler and his Message how toTheorybased According to Tobler we expect attribute values close together to be more similar than those more distant This expectation drives the idea of interpolation and Geostatistical methods A graphical representation of this is the variogram cloud Presentation of the square root of the difference between attribute pairs against the distance between them Variogram Cloudiness Let s look atone example from O Sullivan and Unwin 2003 Might be we understand more about spatial autocorrelation and how to catch it Do we have a trend in here 250 200 150 100 0 50 0 50 100 150 200 250 300 Variogram Cloudiness Can you see a trend If so what does it mean Remember the visual trend in the surface 20 a II Square root of height difference 8 U 1 100 Y 1 1 w 200 300 400 500 Distance between spot heights Variogram Cloudiness This is the representation of NS open circles and EW filled circles directions Which one shows a clearer trend and why How do we call this effect Why are there so few sample points now 5mm m a nInm alumna Variogram Cloudiness For only few points we have lots of pairs to be represented and a variogram or semivariogram becomes harc 5 to be interpreted How to summarize the variogram cloud One way is to use boxplots showing means medians quartiles and extremes for individual lags 2a Regionalized Variable Theory RVT By putting together the last slides you can understand what RVT means The value ofa variable in space at location x can be expressed by summing together three components A structural component eg a spatial trend orjust the mean mx mx Random spatially autocorrelated e39x regionalized variable local spatial W Equot autocorrelation a x Random noise stochastic variation not dependent on location a Zx mx a x a Regionalized Variable Theory RVT Combination of these three components in a mathematical model to develop an estimation function function is applied to measured data to estimate values across area mx is our trend global trend surface a is the random component that has nothing to do with location s x is our regionalized component and describes the local variation of the variable in space This latter relationship can be represented by a dispersion measure Experimental Semivariance here we call it the experimental semivariance for lag h A 1 W 2 m 2N0 mp Zola h Nh is the number of pairs separated by lag h lag tolerance o h lag width or the distance between pairs of points o z is the value of a point at u or uh Remember what we did with the variogram cloud by binning the amount of data points The Experimental Semiviogram Nugget Initial semivariance when autocorrelation is highest intercept orjust the uncertainty where distance is close to 0 2quot Sill Point where the curve levels off inherent variation where there is little autocorrelation Range Lag distance where the sill is reached over which differences are spatially dependent Nugget a Range Stu Lag and Lag Tolerance 2 This has to do with binning 1 i M Lag distance is the direct Kg planar distance between 3 5 r points a and b Tolerances used to define 23 sets of values that are Ia9rolemnce similar distances apart distances rarely repeat in reality Semivariance Covariance and Correlation Spread vs Similarity 39 7 Va39izrce i 3 in D u on 7 39 9 V 1 um i i I Covariance 00 Mh 2 un39zull h mm a V39 a x 7 Correlation 91 x sznante u39ln gamma 1 m BerNaming Squot39an39 117 uh u a mm m ylt MAM H gt A J b a 5 g 9 wr 39 ur J i i i i n my 1191 Emu an i Lag meters from Bohhng 2005 Semivariogram Model The empirical semivariogram allows us to derive a semivariogram model to represent semivariance as a function of separation distance Thus we can infer the characteristics of the underlying process using the model Compute the semivariance between points Interpolate between sample points using an optimal interpolator kriging Constraints for a Semivariogram Model Monotonically increasing Asymptotic max sill Nonnegative intercept nugget Anisotropy 0 A quotK Sill Semivariog ram Models G aussian model 5 Exponential model 5 Sperical model 5 h h h a 2 3 f0 h S 0 C0 exp 0 exp 0 0 010 for h gt L0 339 range C nugget J 5i however fitting the semlvarlogram IS complex What can you think of would happen if there is a trend in the data from httpwwwasu quot 39 39 quot quot Aeer t12htm Where is the Best Fit Porosity Semlvariogram with Three Models l39v xquot Id 1 11 am L Jr metmss from Bohling 2005 Why is Anisotropy Important So far an isotropic structure of spatial correlation has been assumed The semivariogram depends on the lag distance not on direction omnidirectional semivariogram In reality we often have differences in different directions and need an anisotropic semivariogram One model is geometric anisotropy same sill in all direction but within different ranges Geometric Anisotropy Find ranges in three orthogonal principal directions and create a threedimensional lag vector h hxhyhz Transform this 3D vector into an equivalent isotropic lag h Semivariance values computed for pairs that fall within directional bands and prescribed lag limits searching for directional dependenceN A bandwidth Geometric Anisotropy These directional bands are defined by azimuthal direction angular tolerance and bandwidth You did this already in the lab and hopefully knew what you were doing semivariance Lag m Kriging Statisticallybased optimal estimator of spatial variables Predictions based on regionalized variable theory you know this now Kriging first used by Matheron 1963 in honour of DG Krige a south African mining engineer who laid the groundwork for geostatistics Kriging Principles Similar to IDW weights are used with measured variables to estimate values at unknown locations a weighted average Weights are given in a statistically optimal fashion given a specific model assumptions about the trend autocorrelation and stochastic variation in the predicted variable In other words We use the semivariance model to formulate the weights Ordinary Kriging For a basic understanding we look at Ordinary Kriging o Weighting of data points according to distance Estimated values 2 are the sum of a regional mean mx and a spatially correlated random component x mx is implicit in the system Assumptions no trend isotropy variogram can be defined with math model same semivariogram applies for the whole study area variation is a function of distance not location constant mean stationarity of the spatial process 2m where iZZs I 1 Ordinary Krlglng You see it looks similar to IDW But Weight computation is much more complex based on the inverted variogram Weights reach the value zero when we reach the range a Ordinary no trend regional mean is estimated from sample Km A Zsi the measured value at the ith location Sill Iii an unknown weight for the measured value at the ith location 50 the prediction location N the number of measured values Range Nugget 0 How Semivariance Determines Kriging Weights Weighted average of the observed attribute values Weights are a function of the variogram model amp sum to 1 they should to be unbiased You see the sizes of the points proportional to their weights they get Points with distances gt range will have a weight of zero WHY Range2o sm1o NuggeFD Typical Working Steps Describing the data identifying spatial autocorrelation and trends experimental semivariogram n Building the semivariogram model by using the mathematical function Using the semivariogram model f for defining the weights a J Evaluate interpolated surfaces A First Summary Kriging is complex Here you obtained a first overview of the principles of this technique Weights are derived from the semivariogram model to intelligently infere unmeasured values You have seen the underlying rules of the semivariogram and hopefully understood why autocorrelation directional trends and stationarity are important You also understood the difference to other local interpolators don t you

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